%-----------------------------------------------------------------------------
%todo Haui
\chapter{Meaning}
%-----------------------------------------------------------------------------


%-----------------------------------------------------------------------------
\section{Zero Knowledge Proof} 
%-----------------------------------------------------------------------------
"`Usually, a proof of a theorem contains more knowledge than the mere fact that
the theorem is true. \ldots Zero-knowledge proofs are defined as those proofs that
convey no additional knowledge other than the correctness of the proposition in
question"' \cite{goldwasser1989knowledge}. In other words zero knowledge proofs
do proof or falsify a statement of one party - the proofer - through another
party - the verifier - in a way the verifier does not get aware and does not get
to know the statement itself. 
\newline

Therefore the following three properties have to be fulfilled:
\begin{itemize}
  \item Completeness
  \newline 
  if the statement is true, the honest verifier will be convinced of it by an
  honest prover.
  \item Soundness
  \newline  
  if the statement is false, no dishonesty prover can convince the honest
  veri\=fier that it is true, except with some small probability.
  \item Zero knowledge
  \newline  
  if the statement is true, no dishonesty verifier learns anything other than
  this fact. That is what makes it unique. Furthermore replaying eavesdropped
  communication by third parties does not lead to proofed statements
  \cite{kp}.
\end{itemize}

Following this "`a zero-knowledge proof is a proof
that yields nothing but its validity" \cite{Goldreich:1991:PYN:116825.116852}.


\subsection{Abstract Illustration}
To illustrate a zero-knowledge proof we refer to the prover of the statement as
Peggy and to the verifier of the statement as Victor. Let us assume Peggy has
uncovered a secret word used to open a magic door in a cave. The cave is shaped
like a circle, with the entrance on the left side and the magic door blocking on
the other side , as shown in figure ~\ref{Label0}. Victor wants to know whether
\begin{figure}[htbp]
    \begin{minipage}{0.4\textwidth}
     \centering
      \includegraphics[width=0.8\textwidth]{figures/0.png}
      \caption{Illustration of cave}
      \label{Label0}      
    \end{minipage}\hfill
    \begin{minipage}{0.4\textwidth}    
     \centering
      \includegraphics[width=0.8\textwidth]{figures/1.png}
      \caption{Peggy takes eigther way A or B}
      \label{Label-1}      
    \end{minipage}
  \end{figure}
Peggy knows the secret word. However, Peggy does not want to reveal her
knowledge. In fact she only wants to reveal that she knows it. They label the
left and right paths from the entrance A and B. First, Victor waits outside the
cave as Peggy goes in. Peggy takes either path A or B, as shown in figure
~\ref{Label-1}. Victor is not allowed to see which path she takes. Then Victor
enters the cave and shouts the name of the path he wants her to use to return.
Either A or B, chosen at random, as shown in figure ~\ref{Label-2}. Providing
she really knows the magic word she opens the door if necessary and returns along the desired path,
as shown in figure ~\ref{Label-3}.
  \begin{figure}[htbp]
    \begin{minipage}{0.4\textwidth}
    \label{Label-2}
     \centering
      \includegraphics[width=0.8\textwidth]{figures/2.png}
      \caption{Victor communicates desired way}
    \end{minipage}\hfill
    \begin{minipage}{0.4\textwidth}
    \label{Label-3}    
     \centering
      \includegraphics[width=0.8\textwidth]{figures/3.png}
      \caption{Peggy returns on the desired path}
    \end{minipage}
  \end{figure}
However, suppose she did not know the word. She would only be able to return by
the named path if Victor were to give the name of the same path that she had
entered by. Since Victor would choose A or B at random, she would have a 50\%
chance of guessing correctly. If they were to repeat this trick many times, say
20 times in a row, her chance of successfully anticipating all of Victor's
requests would become about one in 1.05 million. Thus, if Peggy repeatedly
appears at the exit Victor names, he can conclude that it is very probable that
Peggy does in fact know the secret word. However, even if Victor is wearing a
hidden camera that records the whole transaction, the only thing the camera will
record is in one case Victor shouting "A!" and Peggy appearing at A or in the
other case Victor shouting "B!" and Peggy appearing at B. A recording of this
type would be trivial for any two people to fake (requiring only that Peggy and
Victor agree beforehand on the sequence of A's and B's that Victor will shout).
Such a recording will certainly never be convincing to anyone but the original
participants. In fact, even a person who was present as an observer at the
original experiment would be unconvinced, since Victor and Peggy might have
orchestrated the whole "`experiment"' from start to finish \cite{kp}
\cite{quisquater1990explain}.



\section{Zero Knowledge Authentication}
Authentication means to verify one persons or instances identity to another
person or instance. In contrast to authorization that is distinct and
subsequent to verify if one person or identity is privileged to carry out
certain actions. \cite[S.111 f]{Needham:2003:AUT:1074100.1074149}. Therefore
zero knowledge authentication means to verify one persons or instance identity
%Haui besser
without revealing any secret information.
















